Aims. The present study aims at providing a deeper insight into the power and limitation of an unsupervised classification algorithm (called Fisher-EM) on spectra of galaxies. This algorithm uses a Gaussian mixture in a discriminative latent subspace. To this end, we investigate the capacity of this algorithm to segregate the physical parameters used to generate mock spectra and the influence of the noise on the classification. Methods. With the code CIGALE and different values for nine input parameters characterising the stellar population, we have simulated a sample of 11 475 optical spectra of galaxies containing 496 monochromatic fluxes. The statistical model and the optimum number of clusters is given in Fisher-EM by the integrated completed likelihood (ICL) criterion. We repeated the analyses several times to assess the robustness of the results. Results. Two distinct classifications can be distinguished in the case of the noiseless spectra. The one above 13 clusters disappears when noise is added, while the classification with 12 clusters is very robust against noise down to a signal to noise ratio (SNR) of 3. At SNR=1, the optimum is 5 clusters, but the classification is still compatible with the previous one. The distribution of the parameters used for the simulation shows an excellent discrimination between classes. A higher dispersion both in the spectra within each class and in the parameter distribution, leads us to conclude that despite a much higher ICL, the classification with more than 13 clusters in the noiseless case is not physically relevant. Conclusions. This study yields two conclusions valid at least for the Fisher-EM algorithm. Firstly, the unsupervised classification of spectra of galaxies is both reliable and robust to noise. Secondly, such analyses are able to extract the useful physical information contained in the spectra and to build highly meaningful classifications. In an epoch of data-driven astrophysics, it is important to trust unsupervised machine learning approaches that do not require training samples which are unavoidably biased.